Help students understand why procedures for computations with fractions make sense (USDE, 2010).
Use contextual tasks.
Explore each operation with a variety of models.
Let estimation and informal methods play a big role in the development of strategies.
Address common misconceptions regarding the operations.
Real-world contexts with plausible numbers.
CCSM is carefully crafted sequence of standards to build understanding (Fennell)
Minilessons for Operations with Fractions, Decimals, and Percents
This book provides number strings to exploring the four operations with fractions, decimals and percents. It draws upon several models including the double open number line, ratio table, and clock. An overview of the book is available for download. This contains some wonderful information!
Adding and Subtracting - Like Denominators
The meaning of addition is the same for fractions and whole numbers.
Addition can be understood as "putting together."
Compose and decompose to add and subtract
Compute sums of whole numbers and fractions by representing the whole number as an equivalent fraction with the same denominator as the fraction.
Convert improper fractions to mixed numbers by decomposing the fraction into a sum of a whole number and a number less than 1. Connect this to finding a quotient with a remainder.
Attend carefully to underlying unit quantities.
3-5 Standards
4.NF.3
4.NBT.6
Math Tool 160
Adding and Subtracting - Unlike Denominators
Begin with sums of fractions with different denominators where one denominator is a divisor of the other, so that only one fraction has to be changed.
Two fractions can be added by subdividing the unit fractions in one using the denominator of the other.
It is not necessary to find a least common denominator. In fact, this may distract from understanding the algorithm.
Solve word problems and estimate to see if the answer makes sense.
3-5 Standards
4.NF.3
5.NF.1
5.NF.2
Who's Right
MT 196
Multiplying and Dividing
4th Grade
Multiply a fraction by a whole number - 4.NF.4
Video - Students create and explain a representation to show multiplying a fraction by a whole number.
5th Grade
Interpret a fraction as division. Solve word problems that have fractional answers. - 5.NF.3
Multiply a whole number by a fraction - 5.NF.4
Multiply a fraction by a fraction - 5.NF.4
Interpret the meaning of multiplication
repeated addition
using models
Thinking of a single fraction as a product
4/5 = 4 x 1/5
The difference between a x b/c and b/c x a
Visuals are different (modeled differently)
Contexts are different
2/3 x 8 = 2/3 of a group of 8
8 x 2/3 = 8 groups of 2/3
Interpreting a/b x b or a/b x a multiple of b
Understanding needs to replace "cancelling"
Interpreting multiplication of fractions using partitioning
Using fraction strips. Focus on modeling.
Interpreting multiplication as a way to describe rectangle areas.
Relate to whole number area model.
Multiplying by n/n gives an equivalent fraction
Interpreting multiplication as scaling
Relative sizes of products
helpful in estimating
x proper = < starting amount
x improper = > starting amount
Prepares students for work in ratios and proportional reasoning
Multiplying mixed numbers
rename as improper fractions
Use the area model
Recognizing multiplication of fraction situations
taking part of something
scaling
calculating area
6th Grade Standard - Multiplying Rational Numbers - focuses on negative numbers
Uncomplicating - Good questions to ask
Van de Walle 94, 105
MT 109
Fact 11
Division
Understand as equal sharing
Dealing appropriately with remainders.
Table of Contents
Real-world contexts with plausible numbers.
CCSM is carefully crafted sequence of standards to build understanding (Fennell)
Minilessons for Operations with Fractions, Decimals, and Percents
This book provides number strings to exploring the four operations with fractions, decimals and percents. It draws upon several models including the double open number line, ratio table, and clock. An overview of the book is available for download. This contains some wonderful information!
Adding and Subtracting - Like Denominators
3-5 Standards
Math Tool 160
Adding and Subtracting - Unlike Denominators
3-5 Standards
Who's Right
MT 196
Multiplying and Dividing
4th Grade
- Multiply a fraction by a whole number - 4.NF.4
- Video - Students create and explain a representation to show multiplying a fraction by a whole number.
5th GradeInterpret the meaning of multiplication
- repeated addition
- using models
Thinking of a single fraction as a product- 4/5 = 4 x 1/5
The difference between a x b/c and b/c x a- Visuals are different (modeled differently)
- Contexts are different
- 2/3 x 8 = 2/3 of a group of 8
- 8 x 2/3 = 8 groups of 2/3
Interpreting a/b x b or a/b x a multiple of b- Understanding needs to replace "cancelling"
Interpreting multiplication of fractions using partitioning- Using fraction strips. Focus on modeling.
Interpreting multiplication as a way to describe rectangle areas.- Relate to whole number area model.
Multiplying by n/n gives an equivalent fractionInterpreting multiplication as scaling
Relative sizes of products
- helpful in estimating
- x proper = < starting amount
- x improper = > starting amount
- Prepares students for work in ratios and proportional reasoning
Multiplying mixed numbers- rename as improper fractions
- Use the area model
Recognizing multiplication of fraction situations- taking part of something
- scaling
- calculating area
6th Grade Standard - Multiplying Rational Numbers - focuses on negative numbersUncomplicating - Good questions to ask
Van de Walle 94, 105
MT 109
Fact 11
Division
Understand as equal sharingDealing appropriately with remainders.